Gap solitons near a band edge of a spatially periodic nonlinear PDE can be formally approximated by solutions of Coupled Mode Equations (CMEs). Here we study this approximation for the case of the 2D Periodic Nonlinear Schrödinger / Gross-Pitaevskii Equation with a non-separable potential of finite contrast. We show that unlike in the case of separable potentials [T. Dohnal, D. Pelinovsky, and G. Schneider, J. Nonlin. Sci. 19, 95-131 (2009)] the CME derivation has to be carried out in Bloch rather than physical coordinates. Using the Lyapunov-Schmidt reduction we then give a rigorous justification of the CMEs as an asymptotic model for reversible non-degenerate gap solitons and even potentials and provide H s estimates for this approximation. The results are confirmed by numerical examples including some new families of CMEs and gap solitons absent for separable potentials.

We investigate second order linear wave equations in periodic media, aiming at the derivation of effective equations in R n , n ∈ {1, 2, 3}. Standard homogenization theory provides, for the limit of a small periodicity length ε > 0, an effective second order wave equation that describes solutions on time intervals [0, T ]. In order to approximate solutions on large time intervals [0, T ε −2 ], one has to use a dispersive, higher order wave equation. In this work, we provide a well-posed, weakly dispersive effective equation and an estimate for errors between the solution of the original heterogeneous problem and the solution of the dispersive wave equation. We use Bloch-wave analysis to identify a family of relevant limit models and introduce an approach to select a well-posed effective model under symmetry assumptions on the periodic structure. The analytical results are confirmed and illustrated by numerical tests.

We address a two-dimensional nonlinear elliptic problem with a finite-amplitude periodic potential. For a class of separable symmetric potentials, we study the bifurcation of the first band gap in the spectrum of the linear Schrödinger operator and the relevant coupled-mode equations to describe this bifurcation. The coupled-mode equations are derived by the rigorous analysis based on the Fourier-Bloch decomposition and the Implicit Function Theorem in the space of bounded continuous functions vanishing at infinity. Persistence of reversible localized solutions, called gap solitons, beyond the coupled-mode equations is proved under a non-degeneracy assumption on the kernel of the linearization operator. Various branches of reversible localized solutions are classified numerically in the framework of the coupled-mode equations and convergence of the approximation error is verified. Error estimates on the time-dependent solutions of the Gross-Pitaevskii equation approximated by solutions of the coupled-mode equations are obtained for a finite-time interval.

We consider the nonlinear curl-curl problem ∇×∇×U +V (x)U = Γ(x)|U | p−1 U in R 3 related to the nonlinear Maxwell equations for monochromatic fields. We search for solutions as minimizers (ground states) of the corresponding energy functional defined on subspaces (defocusing case) or natural constraints (focusing case) of H(curl; R 3 ). Under a cylindrical symmetry assumption on the functions V and Γ the variational problem can be posed in a symmetric subspace of H(curl; R 3 ). For a strongly defocusing case ess sup Γ < 0 with large negative values of Γ at infinity we obtain ground states by the direct minimization method. For the focusing case ess inf Γ > 0 the concentration compactness principle produces ground states under the assumption that zero lies outside the spectrum of the linear operator ∇ × ∇ × +V (x). Examples of cylindrically symmetric functions V are provided for which this holds.

The nonlinear Schrödinger/Gross-Pitaevskii equation with a linear periodic potential and a nonlinearity coefficient Γ with a discontinuity supports stationary localized solitary waves with frequencies inside spectral gaps, so called surface gap solitons (SGSs). We compute families of 1D SGSs using the arclength continuation method for a range of values of the jump in Γ. Using asymptotics, we show that when the frequency parameter converges to the bifurcation gap edge, the size of the allowed jump in Γ converges to 0 for SGSs centered at any xc ∈ R.Linear stability of SGSs is next determined via the numerical Evans function method, in which the stable and unstable manifolds corresponding to the 0 solution of the linearized spectral ODE problem are evolved up to a common location where the determinant of their bases, i.e., the Evans function, is evaluated. Zeros of the Evans function coincide with eigenvalues of the linearized operator. Far from the SGS location the manifolds are spanned by exponentially decaying/increasing Bloch functions. As we show, evolution of the manifolds suffers from stiffness. A numerically stable formulation is possible in the exterior algebra formulation and with the use of Grassmanian preserving ODE integrators. Eigenvalues with a positive real part larger than a small constant are then detected via the use of the complex argument principle and a contour parallel to the imaginary axis. The location of real eigenvalues is found via a straightforward evaluation of the Evans function along the real axis and several complex eigenvalues are located using Müller's method. The numerical Evans function method is described in detail in order to facilitate its use as a practical tool for locating eigenvalues. Our results show the existence of both unstable and stable SGSs (possibly with a weak instability), where stability is obtained even for some SGSs centered in the domain half with the less focusing nonlinearity. Direct simulations of the PDE for selected SGS examples confirm the results of Evans function computations.

We consider the existence of localized modes corresponding to eigenvalues of the periodic Schrödinger operator −∂ 2x + V (x) with an interface. The interface is modeled by a jump either in the value or the derivative of V (x) and, in general, does not correspond to a localized perturbation of the perfectly periodic operator. The periodic potentials on each side of the interface can, moreover, be different. As we show, eigenvalues can only occur in spectral gaps. We pose the eigenvalue problem as a C 1 gluing problem for the fundamental solutions (Bloch functions) of the second order ODEs on each side of the interface. The problem is thus reduced to finding matchings of the ratio functions R ± = ψ ± (0) ψ±(0) , where ψ ± are those Bloch functions that decay on the respective half-lines. These ratio functions are analyzed with the help of the Prüfer transformation. The limit values of R ± at band edges depend on the ordering of Dirichlet and Neumann eigenvalues at gap edges. We show that the ordering can be determined in the first two gaps via variational analysis for potentials satisfying certain monotonicity conditions. Numerical computations of interface eigenvalues are presented to corroborate the analysis.

We analyze a homogenization limit for the linear wave equation of second order. The spatial operator is assumed to be of divergence form with an oscillatory coefficient matrix a ε that is periodic with characteristic length scale ε; no spatial symmetry properties are imposed. Classical homogenization theory allows to describe solutions u ε well by a non-dispersive wave equation on fixed time intervals (0, T ). Instead, when larger time intervals are considered, dispersive effects are observed. In this contribution we present a well-posed weakly dispersive equation with homogeneous coefficients such that its solutions w ε describe u ε well on time intervals (0, T ε −2 ). More precisely, we provide a norm and uniform error estimates of the form u ε (t)−w ε (t) ≤ Cε for t ∈ (0, T ε −2 ). They are accompanied by computable formulas for all coefficients in the effective models. We additionally provide an ε-independent equation of third order that describes dispersion along rays and we present numerical examples.

We consider light propagation in a Kerr-nonlinear 2D waveguide with a Bragg grating in the propagation direction and homogeneous in the transverse direction. Using Newton's iteration method we construct both stationary and travelling solitary wave solutions of the corresponding mathematical model, the 2D nonlinear coupled mode equations (2D CME). We call these solutions 2D gap solitons due to their similarity with the gap solitons of 1D CME (fiber grating). Long-time stable evolution preserving the solitary fashion is demonstrated numerically despite the fact that, as we show, for the 2D CME no local constrained minima of the Hamiltonian functional exist. Building on the 1D study of [1], we demonstrate trapping of slow enough 2D gap solitons at localized defects. We explain the mechanism of trapping as resonant transfer of energy from the soliton to one or more nonlinear defect modes. For a special class of defects, we construct a family of nonlinear defect modes by numerically following a bifurcation curve starting at analytically or numerically known linear defect modes. Compared to 1D the dynamics of trapping are harder to fully analyze and the existence of many defect modes for a given defect potential causes that slow solitons store a part of their energy for virtually all of the studied attractive defects.